Equivariant quantization of orbifolds

Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces... In this work, we prove existence of an equivariant quantization for orbifolds. Our construction combines an appropriate desingularization of any Riemannian orbifold by a foliated smooth manifold, with the foliated equivariant quantization that we built in \cite{PoRaWo}. Further, we suggest definitions of the common geometric objects on orbifolds, which capture the nature of these spaces and guarantee, together with the properties of the mentioned foliated resolution, the needed correspondences between singular objects of the orbifold and the respective foliated objects of its desingularization.Comment: 13 page

Equivariant symbol calculus for differential operators acting on forms

We prove the existence and uniqueness of a projectively equivariant symbol map (in the sense of Lecomte and Ovsienko) for the spaces $D_p$ of differential operators transforming p-forms into functions. These results hold over a smooth manifold endowed with a flat projective structure. As an application, we classify the Vect(M)-equivariant maps from $D_p$ to $D_q$ over any manifold M, recovering and improving earlier results by N. Poncin. This provides the complete answer to a question raised by P. Lecomte about the extension of a certain intrinsic homotopy operator.Comment: 14 page

Projectively equivariant quantizations over the superspace Rp∣q\R^{p|q}

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra pgl(p+1|q) is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical situations. When the projective superalgebra is not simple (i.e. in the case of pgl(n|n)\not\cong sl(n|n)), we show the existence of a one-parameter family of equivariant quantizations. We also provide explicit formulas in terms of a generalized divergence operator acting on supersymmetric tensor fields.Comment: 19 page

Decomposition of symmetric tensor fields in the presence of a flat contact projective structure

Let $M$ be an odd-dimensional Euclidean space endowed with a contact 1-form $\alpha$. We investigate the space of symmetric contravariant tensor fields on $M$ as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up by those vector fields that preserve the contact structure. If we consider symmetric tensor fields with coefficients in tensor densities, the vertical cotangent lift of contact form $\alpha$ is a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symmetric density valued tensor fields. This generalized Hamiltonian operator on the symbol space is invariant with respect to the action of the projective contact algebra $sp(2n+2)$. The preceding invariant operators lead to a decomposition of the symbol space (expect for some critical density weights), which generalizes a splitting proposed by V. Ovsienko

Natural and projectively equivariant quantizations by means of Cartan Connections

The existence of a natural and projectively equivariant quantization in the sense of Lecomte [20] was proved recently by M. Bordemann [4], using the framework of Thomas-Whitehead connections. We give a new proof of existence using the notion of Cartan projective connections and we obtain an explicit formula in terms of these connections. Our method yields the existence of a projectively equivariant quantization if and only if an \sl(m+1,\R)-equivariant quantization exists in the flat situation in the sense of [18], thus solving one of the problems left open by M. Bordemann.Comment: 13 page

On sl(2)-equivariant quantizations

By computing certain cohomology of Vect(M) of smooth vector fields we prove that on 1-dimensional manifolds M there is no quantization map intertwining the action of non-projective embeddings of the Lie algebra sl(2) into the Lie algebra Vect(M). Contrariwise, for projective embeddings sl(2)-equivariant quantization exists.Comment: 09 pages, LaTeX2e, no figures; to appear in Journal of Nonlinear Mathematical Physic

Abundance and Characteristics of Microplastics in Commercially sold fishes from Cebu Island, Philippines

This study documents microplastics (MPs) in the top three commercially sold fishes viz. Auxis rochei, Rastrelliger kanagurta and Chanos chanos in major public markets of Cebu Island, Philippines. MPs were found in the gastrointestinal tracts (FGIT) and quantified and characterized according to size, type, and color. In general, nine (97.3%) of 81 FGIT samples contained 635 total pieces of MPs with size ranging 0.01 to 0.50 and 1.00 to 2.00 mm. Transparent microfibers (91%) were the most predominant MPs, with blue (48%) as the most common MP color observed, followed by red (39%), black (8%) and white (5%). Chanos chanos proved to be the most susceptible fish to MP ingestion with a mean average of 11.6 pieces per individual fish, followed by A. rochei with 6.6 pieces, and R. kanagurta with 5.3 pieces. The results indicated that MPs were ubiquitous and high in commercially sold fishes in major public wet markets of Cebu Island, Philippines. The ingestion of fishes is of primary concern as a route of human exposure to MPs because they filter a large volume of seawater and are typically eaten whole without gut removal. Further study is needed on the potential consequences of MPs to aquatic populations to assess comprehensive exposure integrating multiple sources and routes.

Cervix carcinoma is associated with an up-regulation and nuclear localization of the dual-specificity protein phosphatase VHR

BACKGROUND: The 21-kDa Vaccinia virus VH1-related (VHR) dual-specific protein phosphatase (encoded by the DUSP3 gene) plays a critical role in cell cycle progression and is itself regulated during the cell cycle. We have previously demonstrated using RNA interference that cells lacking VHR arrest in the G1 and G2 phases of the cell cycle and show signs of beginning of cell senescence. METHODS: In this report, we evaluated successfully the expression levels of VHR protein in 62 hysterectomy or conization specimens showing the various (pre) neoplastic cervical epithelial lesions and 35 additional cases of hysterectomy performed for non-cervical pathologies, from patients under 50 years of age. We used a tissue microarray and IHC technique to evaluate the expression of the VHR phosphatase. Immunofluorescence staining under confocal microscopy, Western blotting and RT-PCR methods were used to investigate the localization and expression levels of VHR. RESULTS: We report that VHR is upregulated in (pre) neoplastic lesions (squamous intraepithelial lesions; SILs) of the uterine cervix mainly in high grade SIL (H-SIL) compared to normal exocervix. In the invasive cancer, VHR is also highly expressed with nuclear localization in the majority of cells compared to normal tissue where VHR is always in the cytoplasm. We also report that this phosphatase is highly expressed in several cervix cancer cell lines such as HeLa, SiHa, CaSki, C33 and HT3 compared to primary keratinocytes. The immunofluorescence technique under confocal microscopy shows that VHR has a cytoplasmic localization in primary keratinocytes, while it localizes in both cytoplasm and nucleus of the cancer cell lines investigated. We report that the up-regulation of this phosphatase is mainly due to its post-translational stabilization in the cancer cell lines compared to primary keratinocytes rather than increases in the transcription of DUSP3 locus. CONCLUSION: These results together suggest that VHR can be considered as a new marker for cancer progression in cervix carcinoma and potential new target for anticancer therapy

Association of intronic variants of the KCNAB1 gene with lateral temporal epilepsy.

The KCNAB1 gene is a candidate susceptibility factor for lateral temporal epilepsy (LTE) because of its functional interaction with LGI1, the gene responsible for the autosomal dominant form of LTE. We investigated association between polymorphic variants across the KCNAB1 gene and LTE. The allele and genotype frequencies of 14 KCNAB1 intronic SNPs were determined in 142 Italian LTE patients and 104 healthy controls and statistically evaluated. Single SNP analysis revealed one SNP (rs992353) located near the 3'end of KCNAB1 slightly associated with LTE after multiple testing correction (odds ratio=2.25; 95% confidence interval 1.26-4.04; P=0.0058). Haplotype analysis revealed two haplotypes with frequencies higher in cases than in controls, and these differences were statistically significant after permutation tests (Psim=0.047 and 0.034). One of these haplotypes was shown to confer a high risk for the syndrome (odds ratio=12.24; 95% confidence interval 1.32-113.05) by logistic regression analysis. These results support KCNAB1 as a susceptibility gene for LTE, in agreement with previous studies showing that this gene may alter susceptibility to focal epilepsy